Calculating The Length Of Line Segment ST With Endpoints S(-7 -6) And T(2 4)

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In the realm of coordinate geometry, determining the distance between two points is a fundamental concept. This article delves into the process of calculating the length of a line segment, specifically focusing on the segment ST‾{\overline{ST}} with endpoints S(-7, -6) and T(2, 4). We will explore the underlying principles, the application of the distance formula, and provide a step-by-step solution to this problem. Understanding this concept is crucial for various mathematical and real-world applications, including navigation, engineering, and computer graphics. So, let's embark on this journey to master the art of finding the length of a line segment.

Understanding the Distance Formula

The distance formula is the cornerstone of calculating the length of a line segment in a coordinate plane. It is derived from the Pythagorean theorem, which relates the sides of a right triangle. Imagine a right triangle where the line segment ST‾{\overline{ST}} is the hypotenuse, and the legs are parallel to the x and y axes. The lengths of these legs represent the differences in the x-coordinates and y-coordinates of the endpoints, respectively.

Deriving the Formula

Let's consider two points in a coordinate plane, P(x1,y1){P(x_1, y_1)} and Q(x2,y2){Q(x_2, y_2)}. The horizontal leg of our right triangle has a length of ∣x2−x1∣{|x_2 - x_1|}, and the vertical leg has a length of ∣y2−y1∣{|y_2 - y_1|}. Applying the Pythagorean theorem, we have:

distance2=(x2−x1)2+(y2−y1)2{ distance^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 }

Taking the square root of both sides, we arrive at the distance formula:

distance=(x2−x1)2+(y2−y1)2{ distance = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} }

This formula provides a direct method for calculating the distance between any two points in a coordinate plane. It is a versatile tool that can be applied to various geometric problems.

Applying the Distance Formula

Now, let's apply this formula to our specific problem: finding the length of the line segment ST‾{\overline{ST}} with endpoints S(-7, -6) and T(2, 4). We can identify the coordinates as follows:

  • x1=−7{x_1 = -7}
  • y1=−6{y_1 = -6}
  • x2=2{x_2 = 2}
  • y2=4{y_2 = 4}

Substituting these values into the distance formula, we get:

ST=(2−(−7))2+(4−(−6))2{ ST = \sqrt{(2 - (-7))^2 + (4 - (-6))^2} }

Simplifying the expression inside the square root:

ST=(2+7)2+(4+6)2{ ST = \sqrt{(2 + 7)^2 + (4 + 6)^2} }

ST=(9)2+(10)2{ ST = \sqrt{(9)^2 + (10)^2} }

ST=81+100{ ST = \sqrt{81 + 100} }

ST=181{ ST = \sqrt{181} }

Therefore, the length of the line segment ST‾{\overline{ST}} is 181{\sqrt{181}} units. This value can be approximated to a decimal if needed, but leaving it in radical form provides the exact length.

Step-by-Step Solution: Finding the Length of ST‾{\overline{ST}}

To solidify our understanding, let's break down the process of finding the length of ST‾{\overline{ST}} into a step-by-step solution:

Step 1: Identify the Coordinates

The first step is to clearly identify the coordinates of the endpoints. In our case, we have:

  • S(-7, -6), which means x1=−7{x_1 = -7} and y1=−6{y_1 = -6}
  • T(2, 4), which means x2=2{x_2 = 2} and y2=4{y_2 = 4}

Step 2: Apply the Distance Formula

Next, we substitute these coordinates into the distance formula:

ST=(x2−x1)2+(y2−y1)2{ ST = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} }

ST=(2−(−7))2+(4−(−6))2{ ST = \sqrt{(2 - (-7))^2 + (4 - (-6))^2} }

Step 3: Simplify the Expression

Now, we simplify the expression inside the square root:

ST=(2+7)2+(4+6)2{ ST = \sqrt{(2 + 7)^2 + (4 + 6)^2} }

ST=(9)2+(10)2{ ST = \sqrt{(9)^2 + (10)^2} }

ST=81+100{ ST = \sqrt{81 + 100} }

ST=181{ ST = \sqrt{181} }

Step 4: State the Result

Finally, we state the result. The length of the line segment ST‾{\overline{ST}} is 181{\sqrt{181}} units. As mentioned earlier, this can be approximated to a decimal value if required, but the radical form represents the exact length.

Practical Applications and Importance

The concept of finding the length of a line segment is not just a theoretical exercise; it has numerous practical applications in various fields. Let's explore some of them:

Navigation and Mapping

In navigation, determining distances between locations is crucial for planning routes and estimating travel times. Whether it's calculating the distance between two cities on a map or determining the shortest path for a ship at sea, the distance formula plays a vital role. GPS systems, for instance, rely heavily on distance calculations to provide accurate location and navigation information.

Engineering and Construction

Engineers and architects use distance calculations extensively in designing structures and infrastructure. They need to determine the lengths of structural components, the distances between supports, and the overall dimensions of buildings and bridges. Accurate distance measurements are essential for ensuring the stability and safety of these structures.

Computer Graphics and Game Development

In computer graphics and game development, calculating distances is fundamental for creating realistic simulations and interactions. For example, determining the distance between two objects is necessary for collision detection, pathfinding, and rendering 3D scenes. Game developers use distance calculations to create engaging and immersive gaming experiences.

Physics and Astronomy

Physicists and astronomers use distance calculations to study the motion of objects in space and to understand the relationships between celestial bodies. Determining the distances between stars, planets, and galaxies is crucial for understanding the structure and evolution of the universe.

Real-World Examples

Consider a scenario where you need to hang a picture on a wall. To ensure the picture is centered, you need to measure the distance from the center of the wall to the desired hanging point. This involves calculating the length of a line segment. Another example is in sports, where athletes need to judge distances for throwing, jumping, or running. Understanding distance calculations can enhance performance in various athletic activities.

Common Mistakes and How to Avoid Them

While the distance formula is relatively straightforward, there are some common mistakes that students often make. Let's discuss these mistakes and how to avoid them:

Incorrectly Substituting Coordinates

One common mistake is substituting the coordinates in the wrong order. Remember that the formula is (x2−x1)2+(y2−y1)2{\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}}. Ensure that you subtract the x-coordinates and y-coordinates in the same order. For example, if you start with x2{x_2}, you must also start with y2{y_2} when subtracting the y-coordinates.

Forgetting to Square the Differences

Another mistake is forgetting to square the differences (x2−x1){(x_2 - x_1)} and (y2−y1){(y_2 - y_1)}. The squaring operation is crucial because it eliminates negative signs and ensures that the distances are always positive.

Incorrectly Simplifying the Square Root

Simplifying the square root can also be a source of errors. Make sure to correctly identify perfect square factors within the radicand (the number inside the square root). If the result is not a perfect square, leave it in simplified radical form or approximate it to a decimal.

Sign Errors

Sign errors are common when dealing with negative coordinates. Pay close attention to the signs when substituting the values into the formula and when simplifying the expression. Remember that subtracting a negative number is the same as adding a positive number.

Tips to Avoid Mistakes

  • Write down the formula: Before you start solving a problem, write down the distance formula. This will help you remember the correct steps.
  • Label the coordinates: Clearly label the coordinates as x1{x_1}, y1{y_1}, x2{x_2}, and y2{y_2}. This will help you avoid substituting the values in the wrong order.
  • Show your work: Show each step of your calculation. This will make it easier to identify and correct any mistakes.
  • Double-check your answer: After you have found the answer, double-check your calculations to ensure that you have not made any errors.

Conclusion

In conclusion, finding the length of a line segment is a fundamental concept in coordinate geometry with wide-ranging applications. By understanding the distance formula and following a step-by-step approach, you can accurately calculate the distance between any two points in a coordinate plane. Remember to pay attention to detail, avoid common mistakes, and practice applying the formula to various problems. With a solid grasp of this concept, you'll be well-equipped to tackle more advanced topics in mathematics and related fields. The length of the line segment ST‾{\overline{ST}} with endpoints S(-7, -6) and T(2, 4) is 181{\sqrt{181}} units, a result we achieved through careful application of the distance formula and meticulous calculation. Keep practicing, and you'll master this essential skill in no time!